Chiral Perturbation Theory and the first moments of the Generalized Parton Distriputions in a Nucleon

TL;DR

The paper develops a covariant Baryon Chiral Perturbation Theory framework to compute the first moments of parity-even Generalized Parton Distributions in the nucleon, encoding them in the generalized form factors $A_{2,0}^{v,s}(t)$, $B_{2,0}^{v,s}(t)$, and $C_{2,0}^{v,s}(t)$ for two light flavors. It yields explicit ${ m O}(p^2)$ results for both isovector and isoscalar channels, analyzes quark-mass and momentum-transfer dependences, and compares with lattice QCD data to extract low-energy couplings and assess chiral extrapolations. The isovector sector shows small pion-cloud effects and a curvature arising from an infinite tower of $(m_ extpi/M_0)^n$ terms, while the isoscalar sector reveals interesting mass and $t$-dependent behavior, including spin contributions $J_{u+d}\,\approx\,0.24$ at physical pion mass and a potentially large, negative $C_{2,0}^s(t)$ related to the D-term. The work emphasizes the need for next-order ${ m O}(p^3)$ calculations and more precise lattice data at low $t$ and light quark masses to fully constrain the short-distance couplings and validate the chiral extrapolations presented.

Abstract

We discuss the first moments of the parity-even Generalized Parton Distributions (GPDs) in a nucleon, corresponding to six (generalized) vector form factors. We evaluate these fundamental properties of baryon structure at low energies, utilizing the methods of covariant Chiral Perturbation Theory in the baryon sector (BChPT). Our analysis is performed at leading-one-loop order in BChPT, predicting both the momentum and the quark-mass dependence for the three (generalized) isovector and (generalized) isoscalar form factors, which are currently under investigation in lattice QCD analyses of baryon structure. We also study the limit of vanishing four-momentum transfer where the GPD-moments reduce to the well known moments of Parton Distribution Functions (PDFs). For the isovector moment <x>_{u-d} our BChPT calculation predicts a new mechanism for chiral curvature, connecting the high values for this moment typically found in lattice QCD studies for large quark masses with the smaller value known from phenomenology. Likewise, we analyze the quark-mass dependence of the isoscalar moments in the forward limit and extract the contribution of quarks to the total spin of the nucleon. We close with a first glance at the momentum dependence of the isoscalar C-form factor of the nucleon.

Figures (8)

• Figure 1: Loop diagrams contributing to the first moments of the GPDs of a nucleon at leading-one-loop order in BChPT. The solid and dashed lines represent nucleon and pion propagators respectively. The solid dot denotes a coupling to a tensor field from the ${\cal O}(p^0)$ Lagrangean of Eq.(\ref{['p0']}). Aside from the trivial unity-contribution, wavefunction renormalization to the couplings of the order $p^1$ Lagrangean only start to contribute at next-to-leading one loop order.
• Figure 2: (Isoscalar) tensor field coupling to the pion cloud of the nucleon. This process only starts to contribute at next-to-leading one-loop order in BChPT.
• Figure 3: "Fit I" of the ${\cal O}(p^2)$ result of Eq.(\ref{['A20v0']}) to the (preliminary) LHPC lattice data of ref.LHPC. The corresponding parameters are given in Table \ref{['table']}. Note that the phenomenological value at physical pion mass was not included in the fit. The grey band shown indicates the estimate of possible ${\cal O}(p^3)$ corrections as discussed in the text.
• Figure 4: "Fit II" of the ${\cal O}(p^2)$ BChPT result of Eq.(\ref{['A20v0']}) to the (preliminary) LHPC lattice data of ref.LHPCand to the physical point (solid line). The corresponding fit-parameters are given in table \ref{['table']}. The dashed curve shown corresponds to ${\cal O}(p^2)$ result in the HBChPT truncation (see Eq.(\ref{['xu-dHBChPT']})). The shaded area indicates the region where one does not expect that ChEFT can provide a trustworthy chiral extrapolation function due to the large pion-masses involved, albeit the covariant ${\cal O}(p^2)$ result does not signal any breakdown in this region.
• Figure 5: Quark-mass dependence of the isovector moments $B_{2,0}^v(t=0)$ and $C_{2,0}^v(t=0)$. In $B_{2,0}^v(t=0)$ we have varied the (unknown) chiral limit value $b_{2,0}^v$ between $0$ and $+0.5$, as lattice analyses QCDSF_datSESAMLHPC suggest that this moment has a large positive value. For the chiral limit value $c_{2,0}^v$ of $C_{2,0}^v(t=0)$ we have chosen the value 0, as preliminary lattice QCD analyses suggest that this moment is consistent with zero LHPC. The grey bands shown indicate the size of possible higher order corrections to these ${\cal O}(p^2)$ results.